SAN FRANCISCO—Faced with a chronic housing shortage that is getting worse and fueling social tensions, politicians here want to speed up the pace of new residential construction. But that's a tall order in a city where a daunting approval process and stiff local opposition can turn even a small project into a yearslong battle.

Workers are finding jobs by the bucketful in San Francisco, but housing supply hasn't kept pace. About 4,000 housing units have been added in the past three years, for roughly 10,000 new households,...